Abstract: Let be the number of degree number fields with Galois group and whose discriminant satisfies . Under standard conjectures in diophantine geometry, we show that , and that there are monic, quartic polynomials with integral coefficients of height whose Galois groups are smaller than , confirming a question of Gallagher. Unconditionally we have , and that the -class groups of almost all Abelian cubic fields have size . The proofs depend on counting integral points on elliptic fibrations.